That might work. You got me thinking.
I've contemplated using something like this to hold various Arkham Horror
But I never really found a satisfactory way of doing any of it
You could glue some "shelves" on the outside. I just saw something like this for the Power Grid market - edit found it (PDF)
. You'll have to look around for a picture of the final 3-D version.
For pieces that can roll (like Arkham Horror discs, IIRC), you could have a curved (upward) shelf made of foam or posterboard. Pieces would roll toward the middle, if at all.